This invention relates to the field of grid generation, specifically grid generation for computer modeling of governing relations over a problem space.
Computers can be used to model processes or relations occurring in a problem space. For example, computers can be used to model fluid or heat flow through a problem space such as a chamber, a mechanical part, or a portion of the world. The conventional approach involves the following steps:
1. Represent by governing relations (or governing equations) the process or relations to be modeled.
2. Construct a grid that represents the problem space and segregates it into cells appropriate to the resolution of model desired.
3. Discretize the governing relations to the grid cell boundaries.
4. Represent initial and boundary conditions.
5. Solve the governing relations for the grid cells using the initial and boundary conditions.
6. Iterate if necessary (for example, if the governing relations require iteration for convergence).
The governing relations matching many processes are continuous relations. Discretizing such governing relations can introduce discretization errors. Discretization errors can delay or even prevent convergence, making an accurate model more expensive or even impossible to obtain. They can also compromise the accuracy of the resulting model.
Generating a grid that represents the problem space also incurs substantial cost. Grid generation conventionally requires significant interaction and direction from a human user. Specialized grid generation expertise and complex grid generation tools are typically employed to help generate a grid that represents the problem space and is amenable to computer solution of the governing relations. See, e.g., Thompson et al., Numerical Grid Generationxe2x80x94Foundations and Applications, Elsevier Science Publishing Company, 1985; Carey, Computational Gridsxe2x80x94Generation, Adaptation, and Solution Strategies, Taylor and Francis Publishing Company, 1997; Hoffman and Chiang, Computational Fluid Dynamics for Engineers, Engineering Education System, Wichita, Kans., USA, 1993. Correction of deficiencies in the grid typically requires expert analysis of the model results and another costly application of the grid generation expertise and tools.
The cost and deficiencies associated with current grid generation methods restrict the application of computers to model processes and relations. Accordingly, there is a need for grid generation methods that do not require intensive human user direction and that do not lead to modeling inaccuracies.
The present invention provides a method of grid generation that uses the geometry of the problem space and the governing relations to generate a grid. The method can generate a grid with minimized discretization errors, and with minimal user interaction.
The method of the present invention comprises assigning grid cell locations so that, when the governing relations are discretized using the grid, at least some of the discretization errors are substantially zero. Conventional grid generation is driven by the problem space geometry; grid generation according to the present invention is driven by problem space geometry and by governing relations. The present invention accordingly can provide two significant benefits: more efficient and accurate modeling since discretization errors are minimized, and reduced cost grid generation since less human interaction is required.
As an example, grid generation according to the present invention can be practiced for solving Navier-Stokes fluid flow relations. A grid can be generated so that TE1 discretization errors are zero using the problem space geometry and the Navier-Stokes relations. Further, a refined grid can be generated so that TE2 discretization errors are zero using the problem geometry, the Navier-Stokes relations, and intermediate solutions. Grids appropriate to the Navier-Stokes relations and the problem space can accordingly be generated with little human interaction. Also, refined grids can be generated so that finer resolution is obtained where needed, and coarser resolution is used where sufficient.
Advantages and novel features will become apparent to those skilled in the art upon examination of the following description or may be learned by practice of the invention. The objects and advantages of the invention may be realized and attained by means of the instrumentalities and combinations particularly pointed out in the appended claims.